# 1617-257/Homework Assignment 13

## Doing

Solve all the problems in sections 25-26, but submit only your solutions of problems 4 and 8 in section 25 and problems 5 and 6 in section 26. In addition, solve the following problem, though submit only your solutions of parts d and e:

Problem A. Let ${\displaystyle \alpha \colon \{(u,v)\in {\mathbb {R} }^{2}\colon u^{2}+v^{2}\leq 1\}\to {\mathbb {R} }^{3}}$ be given by ${\displaystyle \alpha (u,v)=\left(u-v,\,u+v,\,2(u^{2}+v^{2})\right)}$. Let ${\displaystyle M}$ be the image of ${\displaystyle \alpha }$.

a. Describe ${\displaystyle M}$.

b. Show that ${\displaystyle M}$ is a manifold.

c. Find the boundary ${\displaystyle \partial M}$ of ${\displaystyle M}$.

d. Find the volume ${\displaystyle V(M)}$ of ${\displaystyle M}$.

e. Find ${\displaystyle \int _{M}zdV}$ (where ${\displaystyle z}$ denotes the third coordinate of ${\displaystyle {\mathbb {R} }^{3}}$).

## Submission

Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

This assignment is due in class on Wednesday February 8 by 2:10PM.

### Important

Please write on your assignment the day of the tutorial when you'd like to pick it up once it is marked (Wednesday or Thursday).